Problem: You have found the following ages (in years) of 6 turtles. Those turtles were randomly selected from the 45 turtles at your local zoo: $ 22,\enspace 21,\enspace 59,\enspace 21,\enspace 73,\enspace 27$ Based on your sample, what is the average age of the turtles? What is the standard deviation? You may round your answers to the nearest tenth.
Because we only have data for a small sample of the 45 turtles, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\overline{x}} = \dfrac{22 + 21 + 59 + 21 + 73 + 27}{{6}} = {37.2\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {231.04} + {262.44} + {475.24} + {262.44} + {1281.64} + {104.04}} {{6 - 1}} $ {s^2} = \dfrac{{2616.84}}{{5}} = {523.37\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{523.37\text{ years}^2}} = {22.9\text{ years}} $ We can estimate that the average turtle at the zoo is 37.2 years old. There is also a standard deviation of 22.9 years.